Econ 33220
Homework 2
Please correct any obvious typos. Explain your reasoning.
Problem 1. Suppose that
Try to solve this as much as possible, using pencil-and-paper.
1. State the characteristic polynomial. (Hint: λI − B is block triangular. Thus, the determinant is the product of the determinants of the diagonal blocks. This is similar to calculating the determinant of a triangular matrix, where the determinant is the product of the diagonal elements. Check your linear algebra texts, if this is unclear.)
2. Find the roots of the characteristic polynomial. (Hint: one root should be obvious. The other two solve a quadratic.) What are their absolute values? Denote the absolute value of the complex roots by ρ.
3. Decompose B = V DV−1 with the following normalization.
(a) D should be constructed so that the last diagonal element is a real number and the first diagonal element has a nonnegative imaginary part.
(b) V should be constructed, so that V (3, 1) = V (3, 2) = 13, V (3, 3) = 1.
(c) Write V−1 by stating the entries V (i, j) for i = 1, 2 and j = 1, 2 as some complex number divided by 26.
4. Decompose B as B = W XW−1, where
for some θ ∈ [0, π/2] and some ρ ∈ R+ and where W should be normalized so that W is lower triangular with 1 for each of its diagonal elements. What is the value of cos(θ) and sin(θ)?
5. For the three impulse vectors aj , equal to zero in all entries except for 1 in entry j (e.g. a2 = [0, 1, 0]'), plot the impulse responses up to the horizon k = 20. Explain what you see.
Problem 2. Download the “cay” data set from Lettau’s website at Berkeley. Read that data into MATLAB or R. Define xt = [ct
, at
, yt
]' (I already used “y” for output). Then do the following
1. Estimate the co-integrating vector by running an OLS regression of “c” on “a”, “y” and a constant and provide your coefficient estimates γ = [γa, γy, γconst].
2. Construct your version of “cay”, call it “mycay” per
mycay = c − γa ∗ a − γy ∗ y − γconst
Thus, the cointegrating vector will be
β = [1, −γa, −γy]'
Plot both “cay” and “mycay” as a function of time. They should be rea-sonably close (though perhaps not exactly identical). (Hint: you need to create a time axis for this. I used the MATLAB command “time = 1952:0.25:2019.501;”).
3. Estimate α = [αc, αa, αy]
0 as well as the constants η = [ηc, ηa, ηy]' by running the three OLS regressions
∆zt = ηz − αzmycayt−1 + uz,t
where z is one of c, a, y, ηz is the intercept and αz is a regression coeffi-cients.
4. Calculate B = I3×3 − α ∗ β'.
5. Note that we have
∆xt = η − α ∗ (β'xt−1 − γconst) + ut
Hence, also calculate the intercept matrix
C = η + αγconst
6. Starting from the last observations xT , forecast [ct
, at
, yt
] for three years recursively from the equation
xt = C + Bxt−1
Also, calculate the “no-error-correction” random walk forecast given re-cursively by
starting from the last observation Plot these forecasts as well as the last five years of your data, one plot for each series (if you can, try to put in a vertical line where the forecast starts and/or distinguish data and forecasts by color or line type). Comment on your results.